This is a framing practicum post. We’ll talk about what bistability is, how to recognize bistability in the wild, and what questions to ask when you find it. Then, we’ll have a challenge to apply the idea.
Today’s challenge: come up with 3 examples of bistability which do not resemble any you’ve seen before. They don’t need to be good, they don’t need to be useful, they just need to be novel (to you).
Expected time: ~15-30 minutes at most, including the Bonus Exercise.
What’s Bistability?
The classic picture of bistability is a marble in a double bowl:
The marble has one stable equilibrium on the left, and another on the right. Although the marble has a whole continuum of possible positions, when left to its own devices for a while it will settle down to one of just two positions.
My own head-canonical examples of bistability come from digital electronics. One is the signal buffer: it turns a sorta-low voltage (like 1 V) and into an unambiguously low voltage (like 0.01 V), or a sorta-high voltage (like 4 V) into an unambiguously high voltage (like 4.99 V). One stable equilibrium is at 5 V, the other is at 0 V, and all other voltages get pushed toward one of those two. This is crucial to building large digital circuits: without buffering, 5 V would decay to 4 V then 3 V as we pass through one gate after another, and eventually we wouldn’t be able to tell whether a voltage is supposed to be high or low.
Another electronic example is the latch, one of the standard low-level memory elements in digital circuits. You can think of a latch sort of like the marble-in-a-double-bowl, but with two extra features:
The state of the marble can be read out. One “bowl” represents “0”, and the other “1”.
An input signal can switch the “marble” from one state to the other.
So, to “write” a bit into the memory element, we push the system into the desired “bowl” (i.e. basin of attraction). It then stays in that state indefinitely, and we can read out the stored bit as many times as we like until it is “overwritten” (i.e. the state is set again).
What To Look For
In general, bistability (and multiple stability) should come to mind whenever an analogue system (i.e. a system with continuous state variables) has discrete behavior. In particular, it’s usually necessary for lossless transmission/storage of discrete information—a system with a single stable equilibrium has no long-term memory, since it always returns to the same state.
Useful Questions To Ask
In the double bowl picture earlier in the post, there’s a hump between the two stable equilibria. The higher the hump, the harder it is to push the marble from one equilibrium to the other. In chemistry, we call that height the “activation energy”—the energy which must be provided to move from one stable state to another. If we want to switch the marble’s state (e.g. to write a bit into memory), we need to provide that activation energy, and a “higher hump” makes it harder/more expensive to set the bit. On the other hand, a higher activation energy makes it less likely that random noise will accidentally push the marble from one side to the other, so a higher-hump memory element can store a bit for longer.
In general, other than the usual equilibrium questions, in a bistable system we usually want to know what’s required to change from one stable equilibrium to another. A few ways this can apply:
We want to set the system into one state or another. So, we need to know what kind of “kick” will do that.
We want to make the system more or less likely to switch state. So, we need to know how to raise or lower the “hump” between states.
We see the system change from one state to another, and we want to know what caused the switch. So, we look for kicks which are large enough to overcome the hump, and ignore smaller noise.
We want to know how long it will take the system to accidentally change states due to random noise—i.e. how long the “memory” is reliable.
The Challenge
Come up with 3 examples of bistability which do not resemble any you’ve seen before. They don’t need to be good, they don’t need to be useful, they just need to be novel (to you).
Any answer must include at least 3 to count, and they must be novel to you. That’s the challenge. We’re here to challenge ourselves, not just review examples we already know.
However, they don’t have to be very good answers or even correct answers. Posting wrong things on the internet is scary, but a very fast way to learn, and I will enforce a high bar for kindness in response-comments. I will personally default to upvoting every complete answer, even if parts of it are wrong, and I encourage others to do the same.
Post your answers inside of spoiler tags. (How do I do that?)
Celebrate others’ answers. This is really important, especially for tougher questions. Sharing exercises in public is a scary experience. I don’t want people to leave this having back-chained the experience “If I go outside my comfort zone, people will look down on me”. So be generous with those upvotes. I certainly will be.
If you comment on someone else’s answers, focus on making exciting, novel ideas work — instead of tearing apart worse ideas. Yes, And is encouraged.
I may remove comments which I deem insufficiently kind, even if I believe they are valuable comments. I want people to feel encouraged to try and fail here, and that means enforcing nicer norms than usual.
If you get stuck, look for:
Systems whose long run behavior can end up a few different possible ways, depending on the initial conditions
Analogue systems with discrete behavior
Long-range transmission or long-term storage of discrete information
Bonus Exercise: for each of your three examples from the challenge, what kinds of “kicks” could cause the system to switch state? What controls how strong the kick needs to be? Suppose you want to switch the state, or want to know what caused a state switch; what kicks could you rule out on the basis that they’re too small? Can you do a Fermi estimate for how often a kick large enough to force a state switch happens due to random noise?
This bonus exercise is great blog-post fodder!
Motivation
Much of the value I get from math is not from detailed calculations or elaborate models, but rather from framing tools: tools which suggest useful questions to ask, approximations to make, what to pay attention to and what to ignore.
Using a framing tool is sort of like using a trigger-action pattern: the hard part is to notice a pattern, a place where a particular tool can apply (the “trigger”). Once we notice the pattern, it suggests certain questions or approximations (the “action”). This challenge is meant to train the trigger-step: we look for novel examples to ingrain the abstract trigger pattern (separate from examples/contexts we already know).
The Bonus Exercise is meant to train the action-step: apply whatever questions/approximations the frame suggests, in order to build the reflex of applying them when we notice bistability.
Hopefully, this will make it easier to notice when a bistability frame can be applied to a new problem you don’t understand in the wild, and to actually use it.
Let’s do this
Bipolar disorder, where the two equilibrium are the depression state and the manic state
Cravings for different restaurants, each a different equilibria: when I order in one, I tend to only want to order there for some time.
Tennis exchange, which is a very complex 3D trajectory of the ball, but end up being classified as a point for one player or the other.
Chess move, which is a complex 3D object being manipulated in complex continuous ways, but result in a discrete state (the position)
Market with much competition, where the equilibria capture who is on top/best, and this tends to be maintained until it is completely switched.
Bonus exercise
Bipolar disorder: the kicks are really hard to know and find, probably the longer the period has lasted the more smaller kicks become able to move the state
Cravings for different restaurants: eating once doesn’t do it, but there’s a bound on how many times I want to eat at the same place. Around it, thinking of another place/food might do the trick.
Tennis exchange: The kick truly depends on how far the trajectory needs to be pushed to be inside or outside, or to reach the racket of the player on this side of the net.
Chess move: the kick is the distance you have to move the piece so that it is in an ambiguous place between two cells, or in another cell.
Market with much competition: the kick is for another competitor to beat the winner.
I really like the chess example. Anything continuous that gets discretized is similar. Like the color of a pixel in a photo, or whether you have crossed the finish line in a race.
Chess example is awesome, I had never of that before. Like, if I set down the piece overlapping the edge of a square, I will want to push it unambiguously into the square.
Recycled goods. They seem to have two states—filled and sealed for sale, and clean, empty and sorted for recycling. The period in between, where they are in the process of being consumed, is often only a tiny fraction of the lifespan of, say, a glass beverage bottle.
Broken windows theory. Under this crime-fighting theory, the existence of “crime breeds crime.” So if you can invest in sprucing up and enforcing the law in a crime-ridden neighborhood, it will tend to become and stay a safe neighborhood. Those that accumulate damage will spiral out of control.
Factors like economic shifts or changes in policing policy might cause the equilibrium to shift from one stable regime to the other.
This suggests an empirical prediction: the majority of neighborhoods will cluster into low-crime or high-crime categories, with relatively few neighborhoods in a zone of moderate crime. Of course, a believer in BWT should specify the size of the zone in which BWT is expected to operate. Will we find crime clustered by city? By neighborhood? By block? By building? If we don’t find crime clustering at the level of the city, should this shake our confidence in BWT? Or is the theory that crime clusters at the level of the neighborhood only?
The Matthew Effect. “For to every one who has will more be given, and he will have abundance; but from him who has not, even what he has will be taken away.”
Most recently, I’ve been exploring this in the context of academic publishing. The distribution of citations has been explained by several models, one of which is the Matthew Effect. Personal crises in a researcher’s life might cause them to shift from a high citation index to a low one. A dramatic success might cause movement in the reverse direction.
Good economic examples in 2 & 3. I find 1 particularly novel—it works surprisingly well as a bistable equilibrium, with the “in recycling bin” equilibrium more stable. If a container is near that state—e.g. it’s mostly empty and sitting around in some random spot—then usually someone will empty it and throw it in the bin.
Most complex eukaryotic organisms are either dead or alive. Yes, they can be sick, which is sort of in between, but sick is still “alive”. In general, going from dead to alive is hard… Going from alive to dead requires disrupting any of several important core sub-equilibria of the living system.
It’s snowing out vs. not. Note: didn’t use raining because “misting” felt like more of an in between edge case than lightly snowing.
A door is either open or closed. Depending on the door, switching from closed to open or open to closed requires applying force and maybe adding some sort of friction device to keep the system in its new state.
(Cheating because I’ve seen this before.) Some natural and designed proteins function as switches with multiple stable states of comparable free energies.
I find 2 particularly interesting, because it matches my experience, but I have no idea what mechanism drives the system into discrete-ish states. Now I think about it, clouds seem related: we often see a “partly cloudy” sky with lots of discrete clouds scattered around and empty space between them, rather than a uniform less-concentrated cloudiness throughout the sky. That suggests bistability in cloud formation. What’s up with that?
Polar caps and glaciers. Albedo change sets a high barrier for new growth when gone.
When getting to know a sect you can be in any social relation to them. At some point you will settle in one of two very distinct states. Entering isn’t too hard. Exiting has a very high ‘activation energy’.
Acquired taste preferences (coffee or tea) seem to be bistable. (I’d guess many habits are)
I’ve thought before about onboarding new users to an app in a similar way to #2.
#3 ties nicely to the “mental mountains” model—in particular, we might guess that psychedelics lower the “hump” between two stable states, so it’s easier to switch tastes/habits.
How would that app work? In what way similar? I am failing to see the part worth emulating in my example.
I will definitely read this. I’ve been trying to find these kinds of preferences in myself for some time.
I mean I’ve thought about it in a similar way—i.e. there’s a state where users are basically “in” the app and use it regularly, and a state where they’re not, similar to how people are “in” a group or not. And there’s an activation energy required to go in either direction—e.g. an app might have an onboarding flow that’s a pain in the ass, or leaving it might require finding a substitute and moving your content over. Though the activation energy to leave an app is a lot lower than the activation energy to leave a sect.
1. When my alarm rings in the morning, I can collapse into one of two possible equilibria: Either I stand up, or I hit snooze and continue to lie in bed for another couple of minutes.
2. In Schrödinger’s cat example, either the atom will decay or not, so the cat will either be dead or not, and it doesn’t make sense to describe it to be in a superposition. (Schrödinger’s cat is terribly often totally misinterpreted by the media and others.) (Admittedly the cat already is in one equilibrium and needs a kick to get into the more stable equilibrium of being dead, so this example is bad.)
3. For a particular type of tech startups, those startups will either become relatively big successes or fail eventually (at least most often).
Bonus:
Actually I need to stand up anyway to put my alarm off, but sometimes I just hit snooze anyway and go back to bed. A kick that get’s me out of bed may be sth like me thinking “oh I have a meeting” or “fuck it, let’s stand up although I hit snooze” or so. A kick that makes me go back to bed after having stood up might be that I feel pretty tiered and decide it is best / most productive to go back and sleep a bit more.
alive->dead: The atom decays. Or you take a big axe and slam it into the box with the cat.
dead->alive: Technology that doesn’t exist yet.
big success → fail: not adopting to technology changes (e.g. Kodak), not being innovative, …
fail → big success: not really possible for the same startup I would say, though the people of the startup might find sth else and become a big success later.
Exercise
Life/Death, for a self-correcting system (typically biology but there are others, e.g. glaciers). If the system is alive and damaged it will roll back into a healthy equilibrium, but if it’s damaged enough it won’t be able to repair. (technically there are more then two equilibria here, e.g. chronic diseases)
Any two-player game with “score” such that once you’re winning it’s easy to stay winning / you have a snowball effect (e.g. chess), and vise versa. (this somewhat subsumes the previous example)
Human memory, some things you’ll forget if you don’t review them, but eventually it sticks where you’ll remember it “forever” without spending any effort on review
Bonus exercise
Forensics, given someone died we look for a “kick” (possibly a literal kick) big enough to get them out of healthy equilibria. Raising the hump is left as an exercise to the reader.
Look for mistakes that caused a winning player to lose their advantage. Can we change the rules of the game to make the barrier higher or lower? (e.g. a comeback mechanic)
How many reviews do you need to get over the hump? What kind of reviews? This system lives in a higher dimensional space, e.g. mnemonic based reviews will be more efficient then brute-force. You could also conceptualize “lowering the hump” as learning mnemonic techniques or generally getting better at learning.
1) As a young man I spent some time jumping back and forth between being single and being in a relationship. Both states felt stable, like two distinct equilibria.
2) Also as a young man I had friends who pursued higher education and friends who didn’t; then there were those who alternated between the two states, once or twice or more. Perhaps not the best example because college is not an ending state but a beginning stage.
3) I’m reminded of the first two lines in “All My Love” by Led Zeppelin: “should I drop out of my farmer life / to chase a feather in the wind”. Both states, farming and being a merry gypsy traveler are states of equilibrium, in their own way.…
All of my examples are similar, I know. The first thing that struck me upon reading the post was the human aspect of this concept, how a person’s life can have more than one equilibrium, depending on your starting position and trajectory. Bonus below:
1) She breaks up with me! Or vice versa. Or I’m walking down Grosvenor Square and a girl catches my eye, with bells on her fingers and rings on her shoes. That may set my ball rolling toward the hypothetical equilibrium of her arms. I might as well try.
2) COVID-19 pushes classes online and a struggling student decides he should take the leap into the professional world instead of the academic.
3) I think Robert Plant was singing about something similar to my answer for number 1).
The speed of something thrown up has two equilibria, falling back down (0) or reaching escape velocity and moving away indefintely.
Bottles spend most of their time either empty or filled.
A problem you understand vs a problem you don’t. Trying to solve a rubics cube through the CFOP or ROUX-method, you will consistently move the cube to the solved state, even if someone messed with it. Trying to solve a rubics cube without ever having done it before, you will probably not move very close to a solved state.
A seesaw on a playground. Either side will come back down again if it’s perturbed a little (given it’s reasonably well oiled etc).
People sitting on it and literally kicking with their feet is enough to kick it into a different state. Random noise due to wind is not strong enough, except maybe in a heavy storm.
The handle of a window (usually). Pointing in one direction when closed and another when open. It will stay in either position unless explicitly moved (which makes it different from a door handle).
The activation energy required to kick it depends on whether it’s oiled or rusty etc.
The ruling party / coalition in a democracy. (In general multistable, but in the US you might call it bistable). One type of perturbation are attempted coups, which will be squashed when not strong enough.
Elections are kicks. But only when the elected party is actually different, which might happen when the voters are dissatisfied with the previous one strongly enough, or the election campaign was just better, etc. Another kick would be a successful coup.
#3 is definitely a useful frame, probably lots of insights to be pumped from that one.